Well, you won't really know without taking gravity readings. Specific gravity is a measure of dissolved solids in a liquid. In the case of brewing, how much sugar is dissolved in your wort. As the yeasts ferment the sugars into alcohol, the gravity readings drop until the yeast finish fermenting. You won't really know when you've reached that point until the gravity readings are consistently the same for 2-3 days in a row. With no hydrometer to read the gravity of the beer, there's no way to really know with any accuracy.
However, in most low gravity beers under a starting gravity of 1.060, typically they'll be finished fermenting somewhere from two to three weeks from brew day. This is provided you're using a vigorous, healthy strain of yeast with adequate oxygen and trace elements in the brewing water. If it were me and I had no hydrometer (they cost about $10), i'd let it ferment a full three weeks, even if the activity slows down and the beer "appears" to be done. Be patient, let it sit, it will benefit from a short resting time of about a week after the beer is done fermenting anyway.
